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GEOMETRY and the PETTIS INTEGRAL1 Pettis Integral. X*Jef Dp = Je X*F Dp for Every X* In

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GEOMETRY and the PETTIS INTEGRAL1 Pettis Integral. X*Jef Dp = Je X*F Dp for Every X* In TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 269, Number 2, February 1982 GEOMETRY AND THE PETTIS INTEGRAL1 BY ROBERT F. GEITZ Abstract. Convex sets involving the range of a vector-valued function are con- structed. These constructions provide a complete characterization of the bounded Pettis integrable functions. In this paper we present a geometrical study of the Pettis integral. Our results show that there is a strong link, previously unsuspected, between the range of a vector-valued function and its properties of measurability and integrability. In the course of this paper we will characterize the functions that are equivalent to strongly measurable functions (Theorem 2.8), the bounded Pettis integrable func- tions on arbitrary finite measure spaces (Theorems 3.2 and 3.4), and the bounded Pettis integrable functions on perfect finite measure spaces (Theorem 4.6). Along the way we will develop new insights into the nature of weak measurability and the Pettis integral. The link between measure theory and geometry in Banach spaces is not new. J. A. Clarkson defined the notion of uniform convexity almost fifty years ago as a condition to insure that functions of bounded variation are differentiable almost everywhere and have Bochner integrable derivatives. Radon-Nikodym theorems for the Bochner integral have been a fruitful source of geometrical properties. Rieffel introduced dentability in 1967 in the course of proving a Radon-Nikodym theorem. Slices and exposed points have similar origins. We are approaching measure theory from a different viewpoint from these authors. Rather than a global condition on the range space, we are looking for conditions on the range of an individual function that will guarantee that function to be integrable or measura- ble. This is in the spirit of the Pettis Measurability Theorem, which characterizes the weakly measurable functions that are strongly measurable in terms of the essential separability of their ranges. 1. Terminology. Let (£2, 2, p) be a finite measure space and let A be a Banach space. A function /( ■) from ñ into A is weakly measurable if the scalar function **/(') is measurable for each x* in the dual space A*. The function/ is Pettis integrable if for each E in 2 there is an element of A, denoted }E f dp, that satisfies x*JEf dp = jE x*f dp for every x* in A*. Received by the editors January 20, 1981. Parts of this paper were presented at the 786th meeting of the Society in Pittsburgh, Pennsylvania on May 16, 1981. 1980 Mathematics Subject Classification Primary 28B05, 46G10. Key words and phrases. Pettis integral, weak measurability, perfect measure, Banach space. 'This material constitutes a portion of the author's Ph.D. thesis from the University of Illinois under the direction of Professor J. J. Uhl, Jr. © 1982 American Mathematical Society 0002-9947/82/0000-0771/$04.50 535 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 536 R. F. GEITZ Another form of measurability can be attributed to some vector-valued func- tions. If/ is almost everywhere the limit (in the norm topology of A) of a sequence of simple functions, then / is strongly measurable. The Pettis Measurability Theo- rem [9, Theorem 1.1] says that a function is strongly measurable if and only if it is weakly measurable and off a null set it has separable range. If / is strongly measurable and if /||/|| dp < oo then the Bochner integral of/exists as an element of A [1, Theorem 2, p. 45], and/is trivially Pettis integrable. The Bochner integral has received considerable study-see [1]. Functions / and g with values in A are said to be equivalent if x*f = x*g a.e. for every x* in A*. Functions that are equivalent to strongly measurable functions need not themselves be strongly measurable. Example 2.3 gives a function / for which x*f = 0 a.e. for each x* in A*, yet/is not strongly measurable. 2. The core. The principal tool in this section, the core, is related to Rieffel's notion of the essential range of a vector-valued function. After developing a few of the technical aspects of this tool, we use it to characterize the functions that are equivalent to strongly measurable functions. Definition 2.1. Let /: ß —>A and let ££l The core of / over E, denoted corj(E), is the subset of A given by the formula cor/F)= H ~cöf(E\A). Our first theorem suggests that core is closely tied to the integrability of vector-valued functions. Theorem 2.2. Iff is a Pettis integrable function from ñ into X, then for each set E in 2 corf(E)= coi -J— fjdp: B g E, p(B) > oj. Proof. Both directions of this equality are valid precisely because they hold for scalar functions. First, let 73 be a subset of E of positive measure, and let A be a null set. It follows easily from the Hahn-Banach theorem that —l— [ fdpGœf(B\A)Gœf(E\ A). p(B) JB Hence co{( p(B))~x jBf dp: B G E, p(B) > 0} c cor/F). For containment in the opposite direction, let x be in cory(E), and let ||x*|| < 1. For any number e > 0 choose a countable partition tt of E and a function a, constant on the sets in tt, such that the inequality |x*/(0 — «(01 < £/4 holds for all t in E. Note that if B is any set in tt with positive measure and if t G B then, we must have the inequality x*™-mLxVdi\<t/2- Let A be the union of the null sets in tt. Then x E cof(E \ A); hence there is a finite convex sum 2 \f(t¡) such that t,■G E \ A and ||x - 2 V(',)ll < e/2- Thus License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GEOMETRY AND THE PETTIS INTEGRAL 537 we have \x*(x)- 2 \*V(/,)| < «A Finally, for each number i let B¡ be the set in it containing t,. Observe that *•<*>-2v^ i/*'* <e. Since e is arbitrary, we see that x*(x) > inf C*{pW//^:5C£'M(i?)>0 It now follows from the Hahn-Banach theorem that x E CO: {(p(B)Yxjjdp: B g E,p(B) > oj. This completes the proof. This theorem has several immediate consequences. For example, if / is weakly equivalent to a strongly measurable function g, then / has nonempty core. To see why, observe that inside any set of positive measure there is a subset B such that gXB is bounded and Bochner integrable. It follows that fxs is Pettis integrable, and coiy (73) ^ 0. Note that if/and g are weakly equivalent Pettis integrable functions, then they have the same core, since this implies jBfdp = JBg dp for each measurable set 73. A later theorem (Theorem 2.6) will show that this does not depend on the integrability of / and g; equivalent functions always have the same core. Before exploring more fully the properties of the core, we present two examples. The first of these is an easy example that illustrates the utility of the core. Example 2.3. A function with nonseparable range whose core consists of a single point. Let /2[0, 1] be the usual space of countably nonzero functions on [0, 1] that are square-summable, under the /2-norm. For each point t in [0, 1] define e, G m 1]by , v [1 if s = t, e'(s) = 10 n if ( s =£^f t. Define/ from [0, 1] into l2[0, 1] by f(t) = e, for each / in [0, 1]. This function has nonseparable range; in fact, the span of its range is dense in the nonseparable space l2[0, 1]. However, since (/2[0, 1])* = /2[0, 1], the function x*/is only counta- bly nonzero for each x* in A*. Hence x*f = 0 a.e. (with respect to Lebesgue measure,) and/is weakly equivalent to the zero function. The core of/over any set of positive measure thus contains only the zero element of /2[0, 1]. The second example is much more complicated. This is Phillips's example [10, Example 10.8] of a bounded weakly measurable function that is not Pettis integra- ble. Example 2.4. A bounded nonintegrable function with empty core. Let lx[0, 1] be the usual space of all bounded real-valued functions on [0, 1], equipped with the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 538 R. F. GEITZ supremum norm. Under the assumption of the continuum hypothesis, Sierpinski constructed a subset B of the unit square [0, 1] X [0, 1] with properties (1) for each number t0 in [0, 1], the set {j: (s, t0) g B} is countable and (2) for each number s0 in [0, 1], the set {/: (s0, t) $ B} is countable. Let (fi, 2, p) be the standard Lebesgue measure space on [0, 1], and define a function/from [0, 1] into IJO, 1] by [f(s)](t) = XB(s,O- Phillips has shown that/is weakly measurable. In fact, for each x* in /^[0, 1], x*f is almost everywhere constant. We now show that the core of / is empty; hence / is not Pettis integrable. For each number t let A, be the set A, = {s: (s, t) G B}. The properties of B imply that H(At) = 0 for each t. Observe that if s & A, then [f(s)](t) = 0.
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